#509181

order	512
nilpotency class	4
isoclinism	isoclinic class of #509179
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#30444
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂²×C₄ ◁ #1757 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

760 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 64
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 4 generators on 64 points:

(1,2,6,11,7,36,33,46,9,13,35,45,8,12,34,44)(3,14,17,40,37,47,63,42,19,48,56,41,18,43,31,10)(4,20,22,57,24,30,58,53,25,29,59,32,23,5,26,28)(15,49,39,38,21,16,52,27,51,54,64,60,50,55,62,61);(1,3,9,19)(2,10,13,42)(4,21,25,50)(5,27,30,61)(6,31,35,63)(7,18,8,37)(11,43,45,47)(12,40,36,41)(14,46,48,44)(15,23,51,24)(16,53,55,28)(17,33,56,34)(20,38,29,60)(22,39,59,64)(26,52,58,62)(32,54,57,49);(1,4)(2,5)(3,15)(6,22)(7,24)(8,23)(9,25)(10,38)(11,28)(12,29)(13,30)(14,16)(17,39)(18,50)(19,51)(20,36)(21,37)(26,34)(27,40)(31,62)(32,44)(33,58)(35,59)(41,61)(42,60)(43,49)(45,53)(46,57)(47,54)(48,55)(52,63)(56,64);(1,5)(2,4)(3,16)(6,32)(7,29)(8,20)(9,30)(10,39)(11,26)(12,24)(13,25)(14,15)(17,38)(18,54)(19,55)(21,43)(22,44)(23,36)(27,31)(28,34)(33,53)(35,57)(37,49)(40,62)(41,52)(42,64)(45,58)(46,59)(47,50)(48,51)(56,60)(61,63)

Polycyclic presentation:

2^9|a^2=e,b^2=i,e^2=hi,f^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=def,f^a=fi,c^b=cg,d^b=de,e^b=eh,g^b=gi,h^b=hi,f^c=fi,g^c=gi,e^d=eh,f^d=fi,h^d=hi

Conjugacy classes & character table

50 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 416377);